Morse and Lyapunov Spectra and Dynamics on Flag Bundles

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In this paper we study characteristic exponents (Morse and Lyapunov vector spectra) associated to continuous flows evolving on principal bundles. The purpose is to relate these exponents to the dynamics on the flag bundles. We work with invariant flows on a general principal bundle whose structure group G is semi-simple or reductive. The objects to be considered are defined intrinsically from G and the principal bundle. Thus our characteristic exponents take values on a vector subspace a of the Lie algebra g of G, while the flag bundles in question have fiber F, a compact homeneous space of G (generalized flag manifolds of G). When G is a linear group a becomes a subspace of diagonal matrices and F a manifold of flags of subspaces. The vector valued spectra are defined via the cocycle over the flag bundle obtained by projection against an Iwasawa decomposition of G. They measure the exponential growth ratio of the flow the same way as the usual one-dimensional spectra measures the growth ratio of linear flows on vector bundles. (Actually the one-dimensional spectrum can be recovered from the vector valued one, via a representation of G, see Section 9.) This is the set up of Kaimanovich [16] to the proof of the multiplicative ergodic theorem of Oseledets. The Morse spectrum set is a concept of growth ratio along chains introduced by Colonius-Kliemann [5], [6] (see also Colonius-Fabri-Johnson [7] for a vector valued version). Each one of these sets depend on a chain component of a flow. In our case the Morse spectra are defined over chain components of the flow on an associated flag bundle. Our purpose here is to relate the geometry of the vector Lyapunov and Morse spectra with the geometry of the chain components on the flag bundle, which were studied in [4] and [22]. We better explain our results with the concrete example where G = Sl(3, R). Let X be a compact Hausdorff space endowed with a continuous flow (t, x) 7→ t · x (t ∈ T = Z or R) and suppose that φt (x, g) = (t · x, ρ (t, x) g) is a flow on Q = X × G with ρ a continuous cocycle taking values in G. Let A ⊂ G be the subgroup of diagonal matrices (positive entries) and N the subgroup of upper triangular unipotent matrices. Then the Iwasawa decomposition of G reads G = KAN with K = SO(3). For x ∈ X and u ∈ K we write ρ (t, x) u = kt (x, u) at (x, u) nt (x, u) ∈ KAN and look at the limit behavior as t →∞ of the matrix a (t, x, u) /t = log at (x, u) /t in the subspace a = log A.

2 This is done in combination with the asymptotics of the compact part kt (x, u). Via the action of G on K ≃ G/AN the component kt (x, u) becomes a flow on K and a (t, x, k) an additive cocycle over this flow. Thus the characteristic exponents defined by a depend on (x, k) ∈ X ×K. Nevertheless it is more convenient to avoid ambiguities and factor out the subgroup M ⊂ K of diagonal matrices with ±1 entries and get a (t, x, b) as a cocycle over the flow φF induced on X × F, where F ≃ K/M ≃ G/MAN is the manifold of complete flags in R3. Now let M be a chain component of the flow φF on X × F. Its associated Morse spectrum set ΛMo (M) ⊂ a is defined by evaluating the cocycle a (t, x, b) on chains in M taking into account the jumps of the chains (see Definition 3.1 below). The Lyapunov spectrum set ΛLy (M) is the set of limits lim a (t, x, b) /t along the trajectories in M. By general facts ΛMo (M) is a compact convex set which contains ΛLy (M), which in turn contains the set of extremal points of ΛMo (M) (see Section 3 below). The chain components in X × F were described in [4], [22] with the as- sumption that the flow on X is chain transitive. The picture is the following: −1 There exists Hφ ∈ a and a continuous map x ∈ X 7→ fφ (x) = gxHφgx , gx ∈ G, into the set of conjugates of Hφ such that each chain component M has the form x∈X {x} × fix (fφ (x)). Here fix(fφ (x)) is a connected component of the fixed point set of the action on F of exp fφ (x) (the connected components canS be labelled accordingly so that they match each other). For example if the diagonal matrix Hφ has distinct eigenvalues λ1 > λ2 > λ3 with eigenvectors e1, e2, e3 then there are 3! = 6 isolated fixed points, namely the flag b0 = (span{e1} ⊂ span{e1, e2}) together with those obtained from b0 by permuting the basic vectors. A similar description holds for fφ (x), x ∈ X, allowing to label the fixed points, as well as the chain components, with the permutation group on three letters S3, which is the Weyl group W for Sl(3, R). In this case there are 6 distinct chain components denoted by Mw with w running over W = S3. The chain components are also labelled by the permutation group even if Hφ has repeated eigenvalues, but now some of the Mw can merge. In our Sl(3, R) example there are four possibilities for Hφ, namely

(a) Hφ = diag(λ1 >λ2 >λ3) with six chain components Mw, w ∈ W.

(b) Hφ = diag(λ1 = λ2 > λ3) with three chain components M1 = M(12), M(23) = M(123) and M(132) = M(13).

3 (c) Hφ = diag(λ1 > λ2 = λ3) with three chain components M1 = M(23), M(12) = M(132) and M(123) = M(13).

(d) Hφ = diag(λ1 = λ2 = λ3) = 0 and the ﬂow is chain transitive on X × F.

(In all the cases M1 is the only attractor component and M(13) the only repeller. The equalities in the second and third cases are due to the cosets Wφw where Wφ is the subgroup of W fixing Hφ.) We say that the matrix Hφ is the block form of φ. It is determined by the dynamics on the flag bundles and is a combination of the parabolic type of the flow and of the reversed flow (see [4], [22] and Section 5 of the present article). The results of this paper provide a similar picture for the Morse and Lyapunov spectra over the chain components in terms of the Weyl group, the Weyl chambers and the block form Hφ of φ. For G = Sl(3, R) the roots of a are the functionals αij (diag(a1, a2, a3)) = ai − aj, for i =6 j ∈{1, 2, 3}. Their kernels are three straight lines containing the nonregular matrices in a, i.e., those having repeated eigenvalues. The complement to the lines is the union of 3! = 6 Weyl chambers where the Weyl group acts simply and transitively (see Figure 1). A chamber is determined by inequalities ai > aj > ak and we denote it by Cw if w ∈ W is the permutation sending (123) to (ijk). The main results of this paper prove the following estimates and symmetry properties of the vector valued Morse and Lyapunov spectra (see Figures 1 and 2):

1. The union w∈W ΛMo (Mw) of the several Morse spectra is invariant under the Weyl group. The same holds for the Lyapunov spectra S w∈W ΛLy (Mw).

2. TheS spectra ΛMo (M1) and ΛLy (M1) of the attractor component are invariant under the subgroup Wφ ﬁxing Hφ. 3. Λ (M ) is contained in the interior of clC . This means that Mo 1 w∈Wφ w the multiplicities of the eigenvalues of any Morse exponent in Λ (M ) S Mo 1 do not exceed those of Hφ. The same estimate holds for ΛLy (M1), since ΛMo (Mw) is the convex closure of ΛLy (Mw).

4. ΛMo (M1) intercepts the closure of every chamber Cw, w ∈ Wφ. Hence there exists a Morse exponent in ΛMo (M1) whose eigenvalues have

4 Figure 1: Morse spectra for ﬂows with G = Sl(3, R): block forms (a) and (b), each spectrum is labeled by its corresponding component.

the same pattern of multiplicities as Hφ.ΛLy (M1) also meets these chambers, but since it is not necessarily convex, it may happen that every Lyapunov exponent has less multiplicities than Hφ.

−1 −1 5. ΛMo (Mw) = w ΛMo (M1) and ΛLy (Mw) = w ΛLy (M1). Com- bining this with the previous statement it follows that distinct Morse spectra do not overlap. (This fact is not true for linear ﬂows on vector bundles as shown in [6], Example 5.5.11. The point here is that the vector bundle Morse spectra are images under linear maps of our spectra, see Section 9. Overlappings of the images may occur.)

The last statement says that the whole Morse spectra can be read off from the spectrum of the attractor component. This phenomenon is already present in the analysis of the chain components in flag bundles, whose main properties are governed by the extremal (attractor and repeller) components. These results show an intimate relation between the spectra and the dynamics on the flag bundles. From the chain components on the bundles we get the symmetries of the spectra. Conversely, from the Morse spectra (of the attractor component only) we can recover the block form Hφ of the flow, and hence its parabolic type. This in turn tells the number of chain components, their geometry, their Conley indices, etc.

5 Figure 2: Morse spectra for ﬂows with G = Sl(3, R): block forms (c) and (d).

The underlying structure behind this relationship is a “block decomposition” of the flow in the following sense: From the chain components on the flag bundles one can build a φt-invariant subbundle Qφ ⊂ Q whose structure group is the centralizer ZHφ of the block form Hφ (see Section 5 below). If R G = Sl(n, ) then ZHφ is a subgroup of block diagonal matrices, hence we say that Qφ is the block reduction of φ. The above results say that the spectra have the same block structure as the block reduction, as expected. In case the bundles Q and Qφ are trivial then the reduction amounts a cohomology of cocycles reducing the original cocycle to one taking values in the subgroup ZHφ . Hence this reduction points towards a Jordan decomposition of the flow, in the sense of Arnold-Cong-Oseledets [1]. A similar picture to the one described in the above examples holds for invariant flows on a general principal bundle whose structure group G is a noncompact semi-simple Lie group with finite center (the Morse spectra for the symplectic group Sp (4, R) are depicted in Figure 3 where a consists of the diagonal matrices diag(λ1,λ2, −λ1, −λ2)). We develop the setup and state our results in the generality of semi-simple Lie groups. These results can be extended to reductive groups (e.g. Gl (n, R)). But now one must add a central component to a and hence to ΛMo. There is not much to say about this central component unless that it is the same for every chain component M.

6 Figure 3: For ﬂows with G = Sp(4, R) the spectrum lives here.

2 Preliminary notation

For the theory of semi-simple Lie groups and their flag manifolds we refer to Duistermat-Kolk-Varadarajan [11], Helgason [14] and Warner [30]. To set notation let G be a noncompact semi-simple Lie group with Lie algebra g. We assume throughout that G has finite center. Fix a Cartan involution θ of g with Cartan decomposition g = k ⊕ s. The form Bθ (X,Y )= −hX,θY i, where h·, ·i is the Cartan-Killing form of g, is an inner product. Choose a maximal abelian subspace a ⊂ s and a Weyl chamber a+ ⊂ a. We let Π be the set of roots of a, Π+ the positive roots corresponding to a+ and Σ the set of simple roots in Π+. The associated Weyl group is W = M ∗/M where M ∗ and M are the normalizer and the centralizer of A, respectively. We write m for the Lie algebra of M. + + The Iwasawa decomposition reads g = k ⊕ a ⊕ n with n = α∈Π+ gα − where gα is the root space associated to α. Put n = α∈−Π+ gα. As to the global decompositions we write G = KS and G = KAN with KP= exp k, S = exp s, A = exp a and N = exp n+. The standardP minimal parabolic subalgebra is defined by p = m ⊕ a ⊕ n+ Associated to a subset of simple roots Θ ⊂ Σ there are several Lie algebras and groups (cf. [30], Section 1.2.4): We write g (Θ) for the (semi-simple) Lie subalgebra generated by gα, α ∈ Θ, and put a (Θ) = g (Θ) ∩ a and n± (Θ) = g (Θ)∩n±. G (Θ) is the connected group with Lie algebra g (Θ) and ± ± A (Θ) = exp a (Θ), N (Θ) = exp n (Θ). Also, aΘ = {H ∈ a : α (H) = 0,

7 α ∈ Θ} and AΘ = exp aΘ. We let ZΘ be the centralizer of aΘ in G and KΘ = ZΘ ∩K. It decomposes as ZΘ = MG(Θ)AΘ which implies that ZΘ = KΘ(S ∩ ZΘ) is a Cartan decomposition while ZΘ = KΘAN(Θ) is an Iwasawa decomposition of ZΘ (which is a reductive Lie group). − The standard parabolic subagebra pΘ = n (Θ)⊕p and the corresponding standard parabolic subgroup PΘ is the normalizer of pΘ in G. The ﬂag manifold of type Θ is FΘ = G/PΘ, which identiﬁes with the set of conjugates of pΘ. The empty set Θ = ∅ gives the minimal parabolic subgroup P = P∅ , which is P = MAN. − We write pΘ = θ(pΘ) for the parabolic subalgebra opposed to pΘ. It is ∗ conjugate to the parabolic subalgebra pΘ∗ where Θ = −(w0)Θ and w0 is the principal involution of W (the one that takes Σ to −Σ). More precisely, − ∗ ∗ − pΘ = w0pΘ if w0 ∈ M is a representative of w0. If PΘ is the parabolic − subgroup associated to pΘ then

− ZΘ = PΘ ∩ PΘ ,

− − and PΘ = NΘ ZΘ. The subset Θ singles out the subgroup of the Weyl group WΘ =(W ∩ PΘ) /M = (W ∩ ZΘ) /M. Alternatively WΘ is the subgroup generated by the reﬂections with respect to the roots α ∈ Θ. For H ∈ a let ΘH = {α ∈ Σ: α(H)=0}. Then we denote the above subalgebras an groups with H instead of ΘH , for example, pH = pΘH , etc.

(cf. Section 5). In this case ZH = ZΘH and KH = KΘH are the centralizer of

H in G and K respectively. Also, WH = WΘH is the subgroup of W ﬁxing H. Conversely, for appropriately chosen elements H ∈ a such that ΘH = Θ, the parabolic subgroups, ﬂags, etc. can be written with reference to H.

3 Morse spectrum of a vector valued cocycle

In this section we discuss the abstract concepts of vector valued cocycles and their Lyapunov and Morse characteristic exponents. We mostly recall the results of Colonius-Kliemann [6] and Colonius-Fabbri-Johnson [7], which are stated in the continuous-time setting for flows on metric spaces. We indicate here the mild modifications needed for discrete-time flows on more general topological spaces. Detailed proofs will be given elsewhere [28].

8 Let E be a compact Hausdorff space and φ : T × E → E a continuous flow with T = R or Z. We denote φ(t, x) = φt(x) = t · x. If V is a finite dimensional normed vector space, a V -valued cocycle over E is a continuous map a : T × E → V with

a(t + s, x)= a(t, s · x)+ a(s, x).

Two V -valued cocycles a and b over E are cohomologous with cohomology h : E → V if a(t, x)+ h(x)= h(t · x)+ b(t, x), where h is a continuous map. A V -valued cocycle a(t, x) over E deﬁnes a ﬂow on E × V by

t · (x, v)=(t · x, a(t, x)+ v).

The product E × V can be viewed as principal V -bundle over E, where the aditive group V acts on the right and leaves invariant the above ﬂow, which is thus made of automorphisms of the bundle. Conversely, let π : P → E be a principal bundle with vector structure group V and φt a ﬂow of automorphisms of P . It is well known that P is a trivial bundle (see e.g. Kobayashi-Nomizu [18]). To each global continuous section χ : E → P there corresponds a trivialization of P and a continuous V -cocycle aχ(t, x) over E given by t · χ(x)= χ(t · x) · aχ(t, x). We note that any two cocycles coming this way from sections are cohomologous. In fact, if η : E → P is another section of P and aη the corresponding cocycle, then there exists a continuous map h : X → V satisfying χ(x)= η(x) · h(x), and it follows that

aη(t, x)+ h(x)= h(t · x)+ aχ(t, x), so that the map h realizes a cohomology between aη and aχ. Vector cocycles coming from such trivializations of principal bundles will show up below. Given x ∈ E, T ∈ T, the ﬁnite-time Lyapunov exponent of the cocycle a at (x, T ) is 1 λ (x)= a(T, x), T T while the Lyapunov exponent of a at (in the direction of) x is

λ(x) = lim λT (x), T →+∞

9 if the limit exists. The Lyapunov spectrum of a subset Y ⊂ E is deﬁned by

ΛLy(Y )= {λ(y) ∈ V : y ∈ Y and λ(y) exists}.

The exponential growth ratio along chains was introduced by Colonius- Kliemann [5], who called them Morse exponents. We use the approach to chains for flows in topological spaces as developed in Patrão [20]. Thus let O be a family of open coverings of the compact Hausdorff space E, which is admissible in the sense of [20]. If U ∈ O is given then two points x, y ∈ E are said to U-close if there exists an open set A ∈ U such that x, y ∈ A. A (U, T )-chain (U ∈ O, T ∈ T) ζ from x to y in E is a finite sequence of points x = x0, x1,...,xN in E and times t0, t1,...,tN−1 in T with ti ≥ T such that ti · xi and xi+1 are U-close for i = 0,...,N − 1. We say that the chain ζ belongs to the subset M ⊂ E if the initial and end points x, y belong to M (but not necessarily the intermediate points). Let a as above be a cocycle over E. Definition 3.1. The finite time Morse exponent λ(ζ) of the (U, T )-chain ζ is 1 N−1 λ(ζ)= a(T , x ), T (ζ) j j j=0 X N−1 where T (ζ) = j=0 Tj is the total time of ζ. We denote by ΛMo(M; U, T ) the set of (U, T )-chains belonging to M and define the Morse spectrum of M to be P ΛMo(M)= {clΛMo(M; U, T ): U ∈ O,T > 0}.

Note that the family\ of coverings O is an ingredient in the definition of ΛMo(M). However this set does not depend on the specific choice of O (see Corollary 3.4 below). It follows immediately from the definitions that

N−1 T λ(ζ)= j λ (x ), T (ζ) Tj j j=0 X so that each ﬁnite-time Morse exponent is a convex combination of ﬁnite-time Lyapunov exponents. That an analogous result is valid for the limit Morse exponents is one of the central results on the structure of Morse exponents which is stated next, together with other results.

10 Theorem 3.2. Let M ⊂ E be a chain component (that is, a maximal chain transitive set).

1. Two cohomologous cocycles have the same Lyapunov and Morse spectra.

2. The Morse spectrum ΛMo(M) of a continuous time ﬂow coincides with any of its discretezations (restriction to cZ, c ∈ R).

3. Suppose that M contains the ω-limit set ω(x) of x ∈ E. Then λ(x) ∈ ΛMo(M), if λ(x) exists.

4. ΛMo(M) is a nonempty compact convex subset of V .

5. ΛMo(M) is the set of integrals q dP with P running through the prob- ability measures supported in M which are invariant by the time-one R ﬂow φ1. Here q : E → V is the time-one map q(x)= a(1, x).

6. The extremal points of ΛMo(M) are Lyapunov exponents so that ΛMo(M) is the closed convex closure of ΛLy(M).

Proof: See [7]. For item (2) and proofs in the present more general context see [28].

Some immediate applications of these results are given in what follows.

Corollary 3.3. Let Mj, j = 1,...,n, be the ﬁnest Morse decomposition of φt in E. Then n

ΛLy(E) ⊂ ΛMo(Mj). j=1 [ Proof: In fact, the Morse components Mj are chain components and contain all the ω-limit sets in E. The result then follows from the above theorem.

Corollary 3.4. The Morse spectrum of a chain component M does not depend on the family of coverings O.

Proof: In fact, by the above theorem we have ΛMo(M) = cl(coΛLy(M)). Since ΛLy(M) does not depend on O the result follows.

11 ′ ′ Corollary 3.5. Let φt be a ﬂow on a compact Hausdorﬀ space E and let b be a V ′-vector cocycle over E′. Suppose that there is a map π : E → E′ and a linear map p : V → V ′ such that the cocycles a and b are related by

b(t, π(x)) = p(a(t, x)), t ∈ T, x ∈ E, and that π(M) is a chain transitive component in E′. Then

ΛMo(π(M), b)= p(ΛMo(M, a)), (1) where the notation indicates which cocycle is being considered. Proof: The Lyapunov exponents of a and b are easily seen to be related by (1) with ΛLy in place of ΛMo. The result then follows by the above theorem.

4 Decompositions of principal bundles and the a-cocycle

Let Q → X be a principal bundle whose structural group G is a reductive Lie group. In this section we build decompositions of Q similar to the Iwasawa and Cartan decompositions of G. We assume throughout that the base space X is paracompact. The right action of G on Q is denoted by q 7→ q ·g, q ∈ Q, g ∈ G.

4.1 Iwasawa decomposition An Iwasawa decomposition G = KAN of the structural group can be carried over to a decomposition of the bundle, which we call the Iwasawa decomposition of Q as well. The construction goes as follows. The group AN is diﬀeomorphic to an Euclidian space and since the base space X of Q is paracompact, by general results on principal bundles there exists a K-reduction of Q, that is, a subbundle R ⊂ Q which is a principal bundle with structural group K (for a proof combine Theorem I.5.7 with Proposition I.5.6 in Kobayashi-Nomizu [18]). In case G is a linear group such reduction can be obtained from a Riemannian metric in an associated vector bundle. For every q ∈ Q there exists g ∈ G and r′ ∈ R such that q = r′ · g. The K-component of the Iwasawa decomposition g = khn ∈ KAN of g leaves

12 R invariant so that, by uniqueness of the Iwasawa decomposition, it follows that every q ∈ Q decomposes uniquely as

q = r · hn, r ∈ R, hn ∈ AN, which exhibits the bundle Q as a product Q = R · AN ≈ R × A × N. We let

R : Q → R, q 7→ r, A : Q → A, q 7→ h, be the associated projections. By the continuity of the Iwasawa decomposition of G and the local triviality of Q, it follows that these projections are continuous. Moreover they satisfy the following properties:

1. R(r)= r, A(r) = 1, when r ∈ R,

2. R(q ·p)= R(q)m, A(q ·p)= A(q)h, when q ∈ Q, p = mhn ∈ P = MAN. In particular, A(r · p)= h.

In what follows we write for q ∈ Q,

a (q) = log A (q) ∈ a.

We use the same notation for the Iwasawa decomposition of g ∈ G, namely, a (g) = log h if g = uhn ∈ KAN. For future reference we note that by the second of the above properties we have

a (q · p)= a (q)+ a (p) , p ∈ P. (2)

Now we discuss the cocycle defined by a. The first step is to get actions on the K-subbundle R. Given an automorphism ϕ ∈ Aut(Q) define the map

ϕR : r ∈ R 7→ R(ϕ(r)) ∈ R.

We have ϕR ◦ ψR =(ϕ ◦ ψ)R R if ψ ∈ Aut(Q). In fact, if ψ (r)= r1 · g, r1 ∈ R, g ∈ AN, then ψ (r)= r1 so that by the above properties

R R R R (ϕ ◦ ψ) (r)= R(ϕ(ψ(r))) = R(ϕ(r1) · g)= R(ϕ(r1)) = ϕ (r1)= ϕ (ψ (r)).

13 We remark that ϕR is not a bundle morphism unless R is ϕ-invariant. Also, when Q = G then ϕR reduces to the usual action of G on K through the Iwasawa decomposition of G. T R Let φt be the continuous flow of automorphisms of Q, t ∈ . Then φt defines a continuous flow in R. We use also the notation a for the cocycle defined by the map a, namely

a T a a R : × R → a, (t, r)= (φt (r)), which is continuous by the continuity of φ and the projection a. It follows from the properties of the Iwasawa decomposition of Q that a is a cocycle R over φs , that is, a a R a (t + s,r)= (t, φs (r)) + (φs,r), where t, s ∈ T, r ∈ R. R In what follows we write simply φt instead of φt .

4.2 a-cocycle over flag bundles The cocycle a over R can be factored to a cocycle over the flag bundle FQ, which is the associated bundle Q ×G F obtained by the left action of G on F. The construction is as follows: Take the closed subgroup MN of G. The quotient space Q/MN can be identified with the associated bundle Q ×G G/MN (see [18], Proposition 5.5). Let P = MAN be the minimal parabolic subgroup, so that the flag bundle FQ ≃ Q/P . Then there is a natural fibration Q/MN → Q/P = FQ, q · MN 7→ q · P , q ∈ Q, whose fiber is P/MN ≃ A. Since MN is normal in P , it follows that this fibration is a principal bundle over FQ with structural group A. This bundle is trivializable (because A is diffeomorphic to an Euclidian space, see [18]). An explicit global cross section is given by a K- reduction R of Q. In fact, any element of FQ can be written r · b0, r ∈ R, where b0 is the origin of F. Consider the map

χ(r · b0)= r · MN ∈ Q/MN.

It is well deﬁned because r1 · b0 = r2 · b0 implies that r2 = r2 · m, m ∈ M, and hence r1 · MN = r2 · MN. It is clearly a cross section of Q/MN → FQ. Its continuity follows from the equality χ(q · b0)= R(q) · MN.

14 Now if φt is a ﬂow of automorphisms of Q, t ∈ T, then the cross section χ deﬁnes a continuous a-valued cocycle a : T × FQ → a by a(t, ξ) = log at where

φt(χ(ξ)) = χ(φt(ξ)) · at, ξ ∈ FQ (cf. Section 3). This is essentially the cocycle over R defined from the Iwasawa decomposition (which justifies our use of the same notation). In fact, let ξ = r · b0 ∈ FQ where r ∈ R and b0 is the origin of F. Take the Iwasawa decomposition φt(r) = rt · atnt ∈ R · AN. Then φt(ξ) = rtb0 so that χ(φt(ξ)) = rt · MN. Also, since nt ∈ N fixes MN, we have φt(χ(ξ)) = rt · atMN = χ(φt(ξ)) · at, in terms of the right action of A on Q/MN. This means that a(t, r · b0) = log at = a(t, r).

4.3 Cartan and polar decompositions Fix a Cartan decomposition G = KS with K as in the Iwasawa decomposition and the corresponding decomposition g = k ⊕ s of the Lie algebra g of G. As before let R be a K-reduction of Q. Then for every q ∈ Q there exists g ∈ G and r′ ∈ R such that q = r′ · g. The K-component of the Cartan decomposition g = ks of g leaves R invariant so that, by the uniqueness of the Cartan decomposition in G, it follows that every q ∈ Q can be written uniquely as q = r · s, r ∈ R, s ∈ S which exhibits the bundle Q as the product Q ≈ R × S. We let RC : Q → R, q 7→ r, S : Q → S, q 7→ s be the associated projections. (We emphasize that the two projections R and RC against the Iwasawa and Cartan decompositions, respectively, are not equal.) By the continuity of the Cartan decomposition of G and the local triviality of Q, it follows that these projections are continuous. Moreover they satisfy the following properties: 1. RC (q · k)= RC (q) · k and S(q · k)= k−1S(q)k if q ∈ Q and k ∈ K. (This is a consequence of the fact that kSk−1 = S.) 2. RC (r)= r and S(r · g)= S(g) if r ∈ R, g ∈ G, where we denote also by S(g) the S-component of g ∈ G = K × S.

15 3. S(q · g) = S(S(q)g) if q ∈ Q and g ∈ G. (In fact, q · g = R(q) · S(q)g so that if S(q)g = kt is the Cartan decomposition of S(q)g in G where t = S(S(q)g) then q · g = R(q) · kt. Hence S(q · g)= t = S(S(q)g).)

Now fix a Weyl chamber A+ sitting inside a maximal abelian A ⊂ S and consider the polar decomposition G = K(clA+)K. If g = ks ∈ KS then g = uhv where u = kv−1 ∈ K and s = v−1hv, h ∈ clA+. Combining this polar decomposition with the Cartan decomposition of Q we can write every q ∈ Q as q = r · hv, r ∈ R, h ∈ clA+, v ∈ K. Here the component h ∈ clA+ is uniquely defined (although r and v are not). Thus we have a well defined map

A+ : Q → clA+, q 7→ h which satisﬁes A+(q · k)= A+(q) if q ∈ Q, k ∈ K. In the sequel we denote with the corresponding lower case letters the logarithms of the above maps:

s(q) = log S(q) ∈ s a+(q) = log A+(q) ∈ cla+.

If one starts instead with a ZΘ-principal bundle QΘ, using the Cartan and Iwasawa decompositions of ZΘ and the same arguments as above one gets the existence of a KΘ-reduction RΘ of QΘ, a Cartan decomposition of q ∈ QΘ given by q = rs, r ∈ RΘ, s ∈ S ∩ ZΘ, and an Iwasawa decomposition of q ∈ QΘ given by

q = rhn, r ∈ RΘ, h ∈ A, n ∈ N(Θ).

Also, if QΘ is a ZΘ-reduction of the G-principal Q, then one can choose a K-reduction R of Q so that RΘ ⊂ R.

4.4 Reductive groups The above cocycles can be also be deﬁned for ﬂows evolving on principal bundles with reductive Lie groups. This is because the Lie algebra g of a reductive Lie group G decomposes as g = g ⊕ z, with g semi-simple and

16 z the center. Hence one can add a central component to a (that is, the split part of z) and get a cocycle on a larger vector space. Also, under natural conditions there is available Cartan and Iwasawa decompositions of G as well as parabolic subgroups and flag manifolds, allowing analogous decompositions of G-principal bundles. We refer to Knapp [17], Chapter VII, for the theory of reductive Lie groups. We will not pursue here the detailed extension of the decompositions for reductive principal bundles. Instead we draw some comments regarding the group Gl(n, R), which show up in the vector bundles (see Section 9). This group is reductive with Iwasawa decomposition O(n, R) AN, where N is the unipotent upper triangular group and A the group of diagonal matrices with positive entries. The Lie algebra a of A decomposes as a = R · id ⊕ a where a is the subspace of zero trace diagonal matrices. Thus the a-valued cocycle a(t, ξ) (constructed the same way as above) writes a(t, ξ)= a0(t, ξ)id+a(t, ξ), ξ ∈ FQ, where a0(t, ξ) is a real valued cocycle which is essentially a trace of a matrix. Actually a0 is constant as a function of ξ because scalar matrices belong to the center of G. Finally the action of Gl (n, R) factors to an action of Sl(n, R) (see [4]), so the dynamics on the flag bundle is the same as for a flow on a Sl(n, R)-bundle.

4.5 Spectra

In the sequel we denote by ΛMo (M) and ΛLy (M) the Morse and Lyapunov vector spectra for the a-cocycle over the flag bundle FQ defined via the trivializable principal bundle Q/MN → FQ. In these expressions M is a chain component of the flow on FQ. For latter reference we note that these spectra does not depend on the trivialization of Q/MN → FQ (and of the K-reduction of Q). This is because two different trivializations yield cohomologous cocycles, which in turn have the same spectra, since the base is compact.

5 Dynamics on ﬂag bundles and block reduction

In this section we describe a φt-invariant subbundle Qφ ⊂ Q → X whose structure group is the centralizer ZHφ of an element Hφ ∈ a (or of its exponential hφ = exp Hφ ∈ A). In case G is a linear group Hφ is a diagonal

17 matrix and the centralizer ZHφ is a subgroup of block diagonal matrices (if the real eigenvalues of Hφ are ordered decreasingly). By analogy to this case we call Qφ the block reduction and Hφ the block form of φt. The invariant subbundle Qφ is constructed after the results of [4], [22] that give the chain recurrent components of φt on the flag bundle FQ. These chain components in turn are modelled fiberwise by the dynamics of the flow on the flag manifold F defined by a one-parameter subgroup exp tY of G. The description of this dynamics can be found [11], Section 3. In the next paragraphs we recall its main features and establish related notation. For the details and proofs see [11]. Let g = k ⊕ s be a Cartan decomposition and a ⊂ s a maximal abelian subspace. Then for every H ∈ a the vector field H induced on F has flow exp tH and is the gradient of a function with respect to a Riemannian metric (Borel metric) in F. Let ZH = {g ∈ G : Ad(g) H =e H} be centralizer of H + in G and put KH = ZH ∩ K. Let a ⊂ a be a Weyl chamber and suppose + + that H ∈ cla . If b0 denotes the origin of F (corresponding to a ) then the connected components of the singularity set of H are the orbits KH ·wb0 with w running through the Weyl group W, which coincide with ZH · wb0. These singularities are the fixed points the one-parametere subgroup exp(tH), t ∈ R. These singularities are transversally hyperbolic submanifolds of F (that is, H has Morse-Bott dynamics) where KH ·b0 is the only attractor and KH ·w0b0 is the only repeller, where w0 is the element of W with largest length (principale −1 involution). Also, KH · w1b0 = KH · w2b0 if and only if w1w2 H = H, that is, if and only if the cosets WH w1 and WH w2 are equal, where WH is the subgroup of W fixing H. In particular, if H ∈ a+ is regular then there are |W| isolated singularities. The stable (respectively unstable) manifold of the component KH · wb0 − − + + is the orbit PH · wb0 = NH KH · wb0 (respectively PH · wb0 = NH KH · wb0) − + of the parabolic subgroup PH (respectively PH ) which is the normalizer of − the parabolic subalgebra pH defined as the sum of the eigenspaces of ad (H) associated with the eigenvalues ≤ 0 (respectively ≥ 0). If Y = Ad(g) H, H ∈ cla+, then g conjugates the induced vector fields

H and Y , so that the singularities of Y are g (KH · wb0) with g (KH · b0) the attractor and g (KH · w0b0) the repeller. Their stable and unstable man- ∓ ifoldse aree respectively g PH · wb0 .e In the sequel we write fix (Y,w) for g (KH · wb0), the w-fixed points of Y , and st(Y,w) for the corresponding − stable manifold g PH · wb0 . 18 Example: If G = Sl(n, R) then F is the manifold of complete flags (V1 ⊂· · · ⊂ n Vn−1) of subspaces Vi ⊂ R with dim Vi = i. We can take a to be the space zero trace of diagonal matrices and a+ the cone of the matrices whose eigenvalues are ordered decreasingly. If H ∈ a+ then H has isolated singularities which are the flags spanned by eigenvalues of H. The attractor is b0 =(he1i⊂he1, e2i⊂··· ), and if w ∈ W is a permutatione then fix(H,w) is + obtained by w-permuting the basic vectors from b0. A nonregular H ∈ cla has repeated eigenvalues which appear in blocks, and ZH is the subgroup of block diagonal matrices, where the blocks have the same size as those of H. KH is the subgroup of orthogonal matrices in ZH . Then it is easy to write down the fixed point components KH · wb0. For instance the attractor fix (H, 1) is made of flags adapted to the block decomposition of H. Let π : Q → X be a principal bundle with semi-simple structural group G and compact Hausdorff base space X. Let φt be a right invariant flow on Q which is chain transitive on X. The flow φt induces a flow in the flag bundle FQ. The following theorem of [4], [22] gives the chain components of this induced flow.

Theorem 5.1. The flow φt on the maximal flag bundle FQ → X admits a finest Morse decomposition, whose Morse components are determined as + follows: There exists Hφ ∈ cla and a φt-invariant map

fφ : Q → Ad(G)Hφ fφ (φt (q)) = fφ (q)

−1 into the adjoint orbit of Hφ, which is equivariant, that is, fφ (q · g)=Ad(g ) fφ (q), q ∈ Q, g ∈ G. The chain components M (w) are parameterized by the Weyl group W and each M(w) is given ﬁberwise as the ﬁxed point set

M(w)π(q) = q · ﬁx(fφ(q),w), q ∈ Q. There is just one attractor component M+ = M(1) and only one repeller − M = M(w0) and their dynamical ordering is the reverse of the algebraic Bruhat-Chevalley order of W.

In this theorem the case Hφ = 0 is not ruled out. In this case the flow is chain transitive on the flag bundles. In a partial flag bundle FΘQ, Θ ⊂ Σ, there exists also the finest Morse decomposition, whose components MΘ (w) are the projections of the components M (w) ⊂ FQ. These projections are also given fiberwise as fixed points of fφ (q), q ∈ Q.

19 Let Θφ = {α ∈ Σ: α (Hφ)=0}. Then the corresponding ﬂag bundle F + Θφ Q has special properties. Namely, the attractor chain component MΘφ =

MΘφ (1) meets each fiber in a singleton, because the attractor fixed point F F component of Hφ on Θφ reduces to a point. The flag Θφ (or the corresponding parabolic subgroup PΘφ , or else Θφ) were called the parabolic type of φ ∗ in [4], [22]. The parabolic type of the reversed flow is the dual Θφ (see [4]) − and in the flag bundle FΘ∗ Q the repeller component M ∗ meets the fibers φ Θφ − in singletons. Furthermore the centralizer ZHφ = PΘφ ∩ PΘφ . In the sequel we say that Hφ is the block form of φ. −1 Now let Qφ = fφ (Hφ) ⊂ Q. Since the adjoint orbit Ad(G) Hφ is identi-

ﬁed to the homogeneous space G/ZHφ , it follow by a well known fact that Qφ is a subbundle of Q with structural group ZHφ (see Kobayashi-Nomizu [18], Proposition I.5.6, where the equivariant map fφ must be viewed as a section of the associated bundle Q ×G G/ZHφ ). The φt-invariance of fφ implies the invariance of Qφ under the ﬂow.

Definition 5.2. If Hφ is the block form of the flow then the ZHφ -subbundle Qφ is called the block reduction of φ. Remark: If Q ≃ X × G is a trivial bundle then the flow is defined by a cocycle ρ (t, x) with values in G: φt (x, g)=(t · x, ρ (t, x) g). By choosing another trivialization of Q the cocycle is changed by cohomologous one. Now the block reduction Qφ is a subbundle of a trivial bundle. It may not be trivial. But if this happens then the original cocycle is cohomologous to a cocycle taking values in the subgroup ZHφ .

Remark: The subgroup ZHφ is algebraic, hence it contains the algebraic hull of the ﬂow (see Feres [12], Zimmer [31]).

The centralizer ZHφ is a reductive Lie subgroup of G. As such it has an

Iwasawa ZHφ = KHφ ANHφ , with compact component KHφ (see Knapp [17],

Chapter VII). Hence, as observed in Section 4.3, the ZHφ -bundle admits a reduction to the compact group KHφ . We denote this subbundle by Rφ. The following statement expresses the chain components on the ﬂag bundles in terms of the block reduction.

Proposition 5.3. For Θ ⊂ Σ, let MΘ(w), w ∈ W, be a chain component of φt on the ﬂag bundle FΘQ. Then

MΘ(w)= {q · wbΘ : q ∈ Qφ} = {r · wbΘ : r ∈ Rφ}, where bΘ is the origin of FΘ.

20 Proof: In fact, if q ∈ Qφ then MΘ(w)π(q) = q · fix(Hφ,w), but fix(Hφ,w)= + ZHφ wbΘ, because Hφ ∈ cla . Then the first equality follows since ZHφ acts transitively on the right in the fiber of q in Qφ. The proof of the second equality is similar, by taking KHφ and Rφ instead of ZHφ and Qφ.

We conclude this section with the construction of a linearization around the attractor of the induced flow in the flag bundle FΘ(φ)Q. This linearization is a flow which evolves on a subbundle of the tangent bundle to the fibers of FΘQ. It will be the main technical tool in the proof of the estimates of Section 7. More details about this construction can be found in the forthcoming paper [23]. We write Θ = Θ(φ)= {α ∈ Σ: α (Hφ)=0} for the parabolic type of φ. As before ZΘ stands for the centralizer of Hφ and Qφ the ZΘ-block reduction. − − Let nΘ be the nilradical of pΘ (which identifies to the tangent space at the F − − origin of Θ). The group ZΘ normalizes nΘ, and hence acts linearly on nΘ by the adjoint representation. Hence, we can build the associated bundle

− VΘ = Qφ ×ZΘ nΘ → X.

Since the ZΘ-action is linear, it follows that VΘ → X is a vector bundle and the ﬂow Φt induced by φt on VΘ is linear. Let bΘ be the origin in FΘ and deﬁne the subset

B − Θ = Qφ · NΘ bΘ and the mapping Ψ : VΘ → BΘ,

− Ψ(q · X)= q · (exp X) bΘ q ∈ Qφ,X ∈ nΘ.

Note that Ψ is well deﬁned because if q′ · X′ = q · X then there exists ′ ′ ′ −1 g ∈ ZΘ with q = q · g so that X = Ad(g) X. But exp X = g exp Xg and ′ ′ gbΘ = bΘ so that q · (exp X ) bΘ coincides with q · (exp X) bΘ. Moreover, Ψ − − is a homeomorphism because the mapping X ∈ nΘ 7→ (exp X) bΘ ∈ NΘ bΘ is a homeomorphism (by Proposition 3.6 of [11]).

Proposition 5.4. The following statements are true.

1. BΘ is an open and dense φt-invariant subset of EΘ which contains the + 0 0 attractor component MΘ =Ψ(VΘ), where VΘ is the zero section of VΘ.

21 2. φt and Φt are conjugate under Ψ:

φt (Ψ(v)) = Ψ(Φt (v)) , v ∈VΘ.

Proof: The φt-invariance of BΘ follows from the φt-invariance of Qφ. To see − F that it is open and dense we note that NΘ bΘ is the open Bruhat cell in Θ, − hence q · NΘ bΘ is open and dense in its fiber. Making q run through Qφ, the result follows. Now, + 0 B MΘ = Qφ · bΘ =Ψ VΘ ⊂ Θ, concluding the proof of the first statement. To see the conjugacy take v = − q · X, q ∈ Qφ, X ∈ nΘ. Then by definition Φt (v)= φt (q) · v, so that

Ψ (Φt (v)) = φt (q) · (exp X) bΘ = φt (q · exp X) bΘ = φt (Ψ(v)) , concluding the proof.

Finally we endow VΘ → X with a natural metric (·, ·) given by

− (r · X, r · Y )= Bθ(X,Y ), r ∈ RΘ,X,Y ∈ nΘ where Bθ (·, ·) is the inner product in the Lie algebra deﬁned by the Cartan involution θ. That this in fact deﬁnes a metric in the whole VΘ follows from − the Iwasawa decomposition Qφ = RφAN(Θ), where AN(Θ) normalizes nΘ.

6 W-invariance of Lyapunov exponents

The purpose of this section is to prove the invariance of the set of a-Lyapunov exponents under the Weyl group (see Corollary 6.8). The proof is based on the concept of regular sequences in a symmetric space (cf. [16]).

6.1 Regular sequences in G We discuss here some asymptotic properties of sequences in G that will be used afterwards in the proof of the invariance of the a-Lyapunov exponents under the Weyl group. Most of the results here are taken from [16], although we need to adapt them to our a-valued exponents. + For a sequence gk ∈ G, k ∈ Z , there are two related limits in a that show up in the Iwasawa or Cartan decompositions of gk.

22 1. The Lyapunov exponent of gk at b ∈ F: 1 λ (gk, b) = lim log a(gku), k→+∞ k

′ where b = ub0, u ∈ G. This limit depends only on b because if b = u b0, ′ ′ ′ u ∈ G, then u = up, for p ∈ P , so by Equation (2) one has a(gku )= a(gku)+ a(p), and in the limit the term a(p) disappears.

2. The polar exponent of gk:

+ 1 + λ (gk) = lim log a (gk), k→+∞ k which depends on the sequence only, contrary to the Lyapunov exponents.

Of course these limits may not exist. Next consider the left coset symmetric space K\G (we use this form instead of G/K to match to the Iwasawa decomposition G = KAN as well as to the right action of G on the principal bundle Q → X). Let d be the G-invariant distance in K\G, which is uniquely determined by

d(x0 · exp(X), x0)= |X|θ, X ∈ s where x0 is the origin of K\G. Following [16] a sequence gk ∈ G is said to be regular if there exists D ∈ s such that d (x0 · gk, x0 · exp kD) has sublinear growth as k → +∞, that is, if 1 d (x · g , x · exp kD) → 0. (3) k 0 k 0

In this case we say that D is the asymptotic ray of gk. (More precisely, x0 ·gk is asymptotic to the geodesic ray x0 · exp tD, t> 0.) If u ∈ K it follows that −1 the sequence gku is also regular with asymptotic ray Ad (u ) D. + Write the polar decomposition of gk as gk = ukhkvk ∈ K (clA ) K. Then one of the main results of [16] says that the sequence is regular if and only if it satisﬁes the following two conditions (see [16], Theorem 2.1):

+ 1 1. The polar exponent λ (gk) = lim k log hk (or equivalently the limit 1/k hk ) exists, and

23 1 2. k d(x0 · gk, x0 · gk+1) → 0, that is, x0 · gk has sublinear growth in K\G. + Moreover, in this case the asymptotic ray of gk is given by D = Ad(u)H + + + where u ∈ K and H = λ (gk) ∈ cla is the polar exponent of gk. The next statement gives the Lyapunov exponents of a regular sequence.

Proposition 6.1. Suppose gk is a regular sequence in G with asymptotic ray D ∈ s. Then the following statements hold.

+ − 1. Suppose D ∈ cla and take y ∈ PD . Then the sequence gky is regular with the same asymptotic ray D. 1 a 2. Suppose D ∈ a. Then limk→+∞ k log (gk)= D.

+ + 3. Let H ∈ cla be the polar exponent of gk. Then the Lyapunov exponent −1 + at b ∈ st(D,w) is given by λ (gk, b)= w H .

Proof: For the ﬁrst statement we note that since the eigenvalues of ad (D) − − on pD are ≤ 0, it follows from the Iwasawa decomposition of y ∈ PD that − the sequence (exp kD) y (exp (−kD)) is bounded in PD so that

d(x0 · (exp kD) y, x0 · exp kD)= d(x0 · (exp kD) y (exp (−kD)) , x0) is bounded as a function of k (cf. [13], Proposition 3.9). But

−1 d(x0·gky, x0·exp kD) ≤ d(x0·gk, x0·exp kD)+d(x0·exp kD, x0·(exp kD) y ).

1 So that k d(x0 · gky, x0 · exp kD) → 0, because gk is asymptotic to exp kD and the last term is bounded. For the second statement write gk = ukaknk ∈ KAN. We have

|log ak − kD|θ = d (x0 · ak, x0 · exp kD) and by a well known inequality of horospherical coordinates (cf. Corollary 1.11 of [19]) the right hand side is bounded above by

d (x0 · aknk, x0 · exp kD)= d (x0 · gk, x0 · exp kD) .

1 Therefore, k log ak → D. − + Now, recall that st(D,w)= uPH+ wb0 where u ∈ K satisﬁes D = Ad(u)H F − and b0 ∈ is the origin. Hence, for b ∈ st(D,w) there exists y ∈ PH+ such

24 1 that b = uywb0. In this case λ (gk, b) = limk→+∞ k log ak where gkuyw = ′ ukaknk ∈ KAN. Since u ∈ K the sequence gku is regular with asymptotic + ray H , so by the ﬁrst statement the same holds for gkuy. But w ∈ K hence −1 + gkuyw is asymptotic to the ray Ad (w ) H ∈ a. Therefore, λ (gk, b) = Ad (w−1) H+, as follows from the second statement, concluding the proof.

In [16] it is also given the following characterization of regularity in terms of the Lyapunov exponents λ (gk, b).

Proposition 6.2. The sequence gk is regular if and only if it has sublinear growth and the Lyapunov exponent λ (gk, b) exists at some (and hence at all) b ∈ F.

Proof: See [16], Theorem 2.5 and its corollary where it is proved that gk is 1 ′ regular it it has sublinear growth and lim k log ak exists, where gk = ukaknk ∈ KAN is the Iwasawa decomposition. But as mentioned above gk is regular if and only if gku, u ∈ K, is regular. Hence the existence of λ (gk, b) at some b ∈ F together with sublinear growth implies regularity.

Now we specialize the above results for sequences gk taking values in a centralizer subgroup ZY , Y ∈ a. By the block reduction dicussed in Section 5 the sequences in G giving rise to our exponents will ultimately belong to a ZY . Proposition 6.3. Take Y ∈ cla+ and put Θ = {α ∈ Σ: α (Y )=0}. Let gk ∈ ZY be a regular sequence in G. Then the asymptotic ray of gk has the form D = Ad(u)H with u ∈ KΘ and H ∈ a such that α(H) ≥ 0 for all α ∈ Θ.

′ Proof: Since ZY = ZΘ we may take the decomposition gk = mkgkhk, with ′ mk ∈ M, gk ∈ G(Θ), hk = exp Hk ∈ AΘ, where Hk ∈ aΘ. Let D be the asymptotic ray of gk so that

′ d(x0 · gk, x0 · exp kD)= d(x0 · gkhk, x0 · exp kD).

′ This implies that gkhk is also regular in G. By Theorem 2.5 of [16] it follows ′ 1 that gk is regular in G(Θ) and the limit k Hk → H exists. Hence, there exists H′ ∈ cla(Θ)+ and u ∈ K(Θ) such that b 1 d′(x′ · g′ , x′ · (exp kH′) u−1) → 0, k 0 k 0 25 where d′ is the distance in the symmetric space K(Θ)\G(Θ) with origin ′ x0 = K(Θ). Since the embeeding K(Θ)\G(Θ) ֒→ K\G is isometric, this ′ limit still holds with the distance d of K\G. Now x0 · gkhk = x0 · gk, and since hk = exp(Hk) ∈ AΘ comutes with u ∈ KΘ and A is abelian we multiply by hk on both arguments of the distance function in the previous limit to conclude that 1 d(x · g , x · exp(kH′ + H )u−1) → 0. k 0 k 0 k Put H = H′ + H. Then the above limits show that

1 ′ −1 b d(x0 · gk, x0 · exp k(H + H)u ) → 0, k so that the asymptotic ray of gk is D = Ad(b u)H. Here u ∈ KΘ and α(H)= α(H′) ≥ 0 for all α ∈ Θ as claimed.

6.2 Lyapunov exponents Let Q = R · AN be an Iwasawa decomposition of the principal bundle Q, where R is a K-reduction of Q. As in Section 4 the logarithm of the A- component gives rise to the a-cocycle a(t, ξ) over the flag bundle FQ and to the a-Lyapunov exponent of the flow φt 1 λ (ξ) = lim a(t, ξ) t→+∞ t in the direction of ξ ∈ FQ. It is useful to write down limits at the bundle level in terms of sequences in the group level and hence to obtain results about the Lyapunov exponents of the flow from the asymptotics in G. The following statement follows directly from the definitions.

Proposition 6.4. Let r ∈ R and write φt(r) = rt · gt, with rt ∈ R, gt ∈ G, t ∈ T. If b0 ∈ F is the origin and b = ub0, u ∈ K, then 1 λ (r · b)= λ(r · ub0) = lim log at, u ∈ G t→+∞ t

′ where gtu = utatnt ∈ KAN is the Iwasawa decomposition.

26 In other words any Lyapunov exponent of the flow is a Lyapunov exponent of a sequence in G. Let Q = R · S ≃ R × S be a Cartan decomposition of the bundle (see Section 4). As before we let S : Q → S stand for the projection onto S ≃ K\G. For any initial condition q ∈ Q we define the sequence S(φk(q)) ∈ S, k ∈ Z+. In what follows we say that q ∈ Q is a regular point for the flow in case S(φk(q)) is a regular sequence in S ⊂ G in the sense of last subsection. We note that q is a regular point if and only if q · g is regular for every g ∈ G. That is, regularity depends only on the fiber of q. In this case we say that the base point x = π (q) ∈ X is regular for the flow. We intend to show that every a-Lyapunov exponent of the flow φt is a Lyapunov exponent of some regular sequence S(φk(q)). To this purpose we check first that as a consequence of continuity combined with compactness of the base space X the sublinear growth of S(φk(q)) always holds. 1 Proposition 6.5. For any q ∈ Q we have k d(x0 · sk, x0 · sk+1) → 0 where sk = S(φk(q)).

Proof: Write qk = φk(q) ∈ Q and let qk = rk · sk ∈ R · S be its Cartan decomposition. Clearly qk+1 = φ1(qk) so that

qk+1 = rk+1 · sk+1 = φ1(qk)= φ1(rk) · sk, by right invariance. Therefore, sk+1 = S(qk+1)= S(φ1(rk)·sk)= S(S(φ1(rk))sk) (see Section 4). Hence

d(x0 · sk, x0 · sk+1)= d(x0 · sk, x0 · S(φ1(rk))sk)=

= d(x0, x0 · S(φ1(rk))) = | log S(φ1(rk))|. By compactness of the base space we have that R is compact. From the continuity of φt and of the Cartan decomposition it follows then that d(x0 · sk, x0 · sk+1) is bounded, which establishes the result.

Now we can prove that any Lyapunov exponent of the ﬂow on Q comes from a regular sequence in G. Proposition 6.6. Let r ∈ R. Then the following statements are equivalent. 1. r is a regular point for the ﬂow. Denote by D ∈ s the asymptotic ray + + and by H ∈ cla the polar exponent of S(φk(r)).

27 2. The Lyapunov exponent λ(r·b) exists in one, and hence in any direction r · b, b ∈ F, along the ﬁber of r. In that case λ(r · b)= w−1H+ for any b ∈ st(D,w).

Proof: Take the Cartan decomposition φk(r) = rk · sk ∈ R · S. Then sk = S(φk(r)) has sublinear growth. By Proposition 6.4 the Lyapunov exponent at r·b, b ∈ F, is the Lyapunov exponent at b of the sequence sk. Now the result follows by Proposition 6.2 which ensures that a sequence with sublinear growth is regular if and only if some (and hence all) Lyapunov exponent exists.

It follows immediately that any Lyapunov exponent of the ﬂow is the Lyapunov exponent of a regular sequence. Corollary 6.7. Take ξ = r · b ∈ FQ and assume that its Lyapunov exponent λ (ξ) under φt exists. Then r is a regular point for the ﬂow and λ (ξ) = λ (sk, b) where sk = S(φk(r)).

By Proposition 6.1 the set of a-Lyapunov exponents λ (gk, b), b ∈ F, of a regular sequence gk is invariant under the Weyl group W. Therefore the above corollary has as a consequence the following symmetry of the Lyapunov spectrum. Corollary 6.8. Take ξ = r · b ∈ FQ and assume that its Lyapunov exponent λ (ξ) for the flow φt exists. Then for every w ∈ W, wλ (ξ) is also an a- Lyapunov exponent of the flow. In other words this last result says that the whole set of Lyapunov exponents is invariant under the Weyl group. Since the Morse exponents are convex combinations of Lyapunov exponents, it follows that the same result holds for the whole Morse spectrum of the flow. This last statement will be clarified later with the description of the Morse spectrum of each chain component. Remark: The Iwasawa decomposition of Q gives an additive cocycle over FQ. The polar decomposition of Q can be shown to give a subaditive a- + + cocycle over X in the following way. Put a (t, x) = a (φt(r)), where r ∈ R and π(r)= x. Let µ ∈ a∗ be a weight of a finite dimensional representation of G, then

+ + + hµ, a (t + s, x)i≤hµ, a (t, φs(x))i + hµ, a (s, x)i.

28 As usual (see [16]) an application of the subadditive ergodic theorem to this cocycle shows that almost all points (with respect to an invariant measure in X) are regular, thus yielding the Multiplicative Ergodic Theorem for the a-Lyapunov exponents. Since these matters are exhaustively treated in the literature we do not exploit it any further.

7 Spectrum of the attractor component and Weyl chambers

In this section we prove one of the main results of this paper. It locates the Morse spectrum of the attractor chain component M+ ⊂ FQ within an open cone in a. This cone is read off from the parabolic type of the flow (see Corollary 7.6). The proof is based on estimates of the linear flow on the vector bundle VΘ(φ) → X discussed at the end of Section 5. We make use of the following statement about linear flows on vector bundles.

Proposition 7.1. Let Φt be a linear ﬂow on a vector bundle V → X with norm |·| over a compact Hausdorﬀ base space. Suppose that the zero section V0 is an attractor. Then there are positive constants C,µ > 0 such that

−µt kΦtkx ≤ Ce , t ≥ 0, x ∈ X, where k·k denotes the operator norm of |·|.

Proof: Let U be an attracting neighborhood of V0 in V. Let v ∈ V be arbitrary. Then εv ∈ U for suﬃciently small ε > 0 which implies that |Φt(εv)| = ε|Φt(v)|→ 0 when t → +∞. Hence, |Φt(v)|→ 0, when t → +∞. The result then follows by the uniformity lemma of Fenichel [12] (see also [6], Lemma 5.2.7, and [25], Theorem 2.7).

The constant µ > 0 is called a contraction exponent of the linear ﬂow. By compactness of the base space it is easily seen to be independent of the chosen metric in V. We have the following estimate of the values of the roots on the a- + Lyapunov exponent in ΛLy(M ).

29 Theorem 7.2. Let µ > 0 be a contraction exponent of the linear ﬂow Φt on VΘ(φ) given in Proposition 5.4. Then every a-Lyapunov exponent λ ∈ + ΛLy(M ) satisﬁes α(λ) ≥ µ, α ∈ Π+ \hΘ(φ)i. (4)

Proof: The main point in the proof is to relate the operator norm of Φt with the cocycle a(t, ξ) over the ﬂag bundle. Let Θ = Θ(φ). Take ξ ∈ M+ and write ξ = r · b0, r ∈ Rφ. Then for t ∈ T we have the Iwasawa decomposition + φt(r)= rt · atnt ∈ RΘAN (Θ) and

a(t, ξ) = log at.

− On the other hand if v = r · Y ∈ VΘ with Y ∈ nΘ then |v| = |Y |θ. But Φt(v)= φt(r) · Y and hence

|Φt(v)| = |φt(r) · Y | = |rt · Ad(atnt)Y | = |Ad(atnt)Y |θ .

Now Ad(at)|n− is positive deﬁnite so that kAd(at)|n− kθ equals its greatest Θ Θ eigenvalue. Since the eigenvalues are e−α(log at), α ∈ Π+ \hΘi, it follows that

+ log kΦtkx ≥ −α(log at)= −α(a(t, ξ)), α ∈ Π \hΘi.

By Proposition 7.1 there exists B ∈ R such that α(a(t, ξ)) ≥ µt + B for all roots α ∈ Π+ \hΘi and t ≥ 0. + 1 a Finally, if λ ∈ ΛLy(M ) then λ = λ(ξ) = limt→∞ t (t, ξ) for some ξ ∈M+, so that α (λ) ≥ µ, concluding the proof.

+ + Corollary 7.3. Take λ ∈ ΛLy (M ) and write Θ(λ)= {α ∈ Π : α (λ)=0}. Then Θ(λ) is contained in the parabolic type Θ(φ).

+ + Since the Morse spectrum ΛMo (M ) is the convex closure of ΛLy (M ) it follows at once that the same estimate of the above theorem holds for the Morse exponents.

+ + Corollary 7.4. If λ ∈ ΛMo (M ) then α(λ) >µ> 0 for every α ∈ Π \ hΘ(φ)i.

30 Now we have the following lemma on root systems, which might be well known. By the lack of a reference we present a proof of it.

Lemma 7.5. The set {H ∈ a : α (H) > 0,α ∈ Π+ \hΘi} is the open convex + + cone int (WΘcla ). Also, two cones int (wiWΘcla ), w1,w2 ∈ W, are either equal or disjoint.

Proof: Let CΘ be the cone of those H ∈ a such that α (H) = 0 if α ∈ Θ and + + β (H) > 0 if β ∈ Π \hΘi. Then β ∈ Π \hΘi if and only if β (CΘ) > 0 (see Warner [30], Lemma 1.2.4.1). Also w ∈ WΘ if and only if wH = H, H ∈ CΘ + and the chambers meeting CΘ in their closures are exactly wa , w ∈ WΘ. Since roots do not change signs on chambers, it follows that α ∈ Π+ \hΘi is + ≥ 0 on WΘcla . + Conversely we check that wcla , w ∈ WΘ, are the only closures of cham- + bers where Π \hΘi is ≥ 0. In fact, take w∈ / WΘ and H ∈ CΘ. Then + wH∈ / WΘcla , because otherwise w would map H to a chamber containing H in its closure and this can happen only if w fixes H, that is only if + w ∈ WΘ. Now, the the set of roots that are > 0 on wH is w (Π \hΘi). By [30], Theorem 1.2.4.8 (and its proof) w (Π+ \hΘi) is not contained in + − hΘi ∪ Π if w∈ / WΘ. Hence there exists γ ∈ Π \hΘi such that γ (wH) > 0, that is, (−γ)(wH) < 0 so that Π+ \hΘi is not positive on wa+. + + So far we have {H ∈ a : α (H) ≥ 0,α ∈ Π \hΘi} = WΘcla . But to take strict inequality > 0 in the left hand side amounts to take interior in the right hand side. −1 To prove the last statement we apply w2 to both cones and reduce to + + the case where w2 = 1. Now if int (w1WΘcla ) meets int (WΘcla ) then, ′ ′ by the first part, they meet at a regular element, say w1wH = w H where ′ ′ + w,w ∈ WΘ and H,H ∈ a are regular. Since W acts simply on the Weyl + ′ ′ −1 chamber a it follows that H = H so that (w ) w1w = 1 since it fixes a regular element. This implies that w1 ∈ WΘ, so that both cones coincide.

By the above lemma we can restate Corollary 7.4 in geometric terms as follows.

+ + Corollary 7.6. ΛMo(M ) is contained in the open convex cone int (Wφcla ).

31 8 Morse spectra and block form

In this section we combine the previous results to get the full picture of the a-Morse spectra of the several chain components M (w) with w running through the Weyl group W. As always we assume that the ﬂow on the base space X is chain recurrent and let Θ = Θ(φ) be the parabolic type of the ﬂow φt. It was proved before that the whole set of a-Lyapunov exponents is W- invariant (see Corollary 6.8). This result will be improved now by showing that the Lyapunov spectra of the chain components are permuted to each other by the Weyl group. To this end we recall the ZΘ-block reduction Qφ of Section 5 as well as the Kφ-reduction Rφ. If b0 stands for the origin in F then by Proposition 5.3 we have

M(w)= Qφ · wb0 = RΘ · wb0.

Lemma 8.1. Let r ∈ RΘ be regular. Then it has asymptotic ray D = + + + Ad (u) H where u ∈ KΘ and H ∈ cla is the polar exponent of r. More- over, Θ(H+) ⊂ Θ.

Proof: Since φk(r) ∈ Qφ we decompose φk (r)= rk ·sk ∈ RΘ ·(S ∩ ZΘ), then sk = S(φk(r)) ∈ ZΘ is regular with asymptotic ray D ∈ s. By Proposition 6.3 we have D = Ad(u) H where u ∈ KΘ and H ∈ a is such that α(H) ≥ 0 for all α ∈ Θ. −1 Noting that S(φk(r · u)) = u sku is a regular sequence in G with asymp- −1 totic ray H = Ad(u ) D it follows that r · u ∈ RΘ is a regular point of the ﬂow with asymptotic ray H ∈ a. Therefore we have the Lyapunov exponent λ(ru · b0) = H (see Proposition 6.6 and Proposition 6.1 (2)). Since + + ru · b0 ∈M it follows that H ∈ ΛLy(M ) so that, applying the estimates of last section, we conclude that α(H) > 0 for all α ∈ Π+ \hΘi (see Corollary 7.3). It follows that Θ (H) ⊂ Θ. Also, α(H) ≥ 0 for all α ∈ Π+ so that H ∈ cla+. This implies that H is the polar exponent of r which is denoted by H+ = H in the statement of the result.

Proposition 8.2. Let r ∈ RΘ be regular with asymptotic ray D ∈ s and polar exponent H+ ∈ cla+. Then for x = π(r) we have

r · fix(D,sw) ⊂M(w)x, s∈W [Θφ 32 −1 + ΛLy(M(w)x)= w WΘφ H . + Proof: By the previous lemma we have D = Ad(u) H with u ∈ KΘ, + + + + H ∈ cla and Θ(H ) ⊂ Θ. Let Hφ be as in Theorem 5.1 so that Θ(H ) ⊂ + Θ = Θ(Hφ). Hence for any w ∈ W, s ∈ WΘ, we have fix (H ,sw) ⊂ fix (Hφ,sw)=fix(Hφ,w), so that

+ r · fix(D,sw)= ru · fix(H ,sw) ⊂ ru · fix(Hφ,w)= M(w). By Proposition 6.1 (3) the above inclusion implies that

−1 + w WΘH ⊂ ΛLy(M(w)x). (5)

Now let ξ ∈M(w)x and write ξ = r·lwb0, l ∈ KΘ. By Proposition 6.1 (3) we −1 + have λ(ξ)= s H where s ∈ W is such that lwb0 ∈ st(D,s). Since u ∈ KΘ we have − − lwb0 ∈ st(X,s)= uPΘ(H+)sb0 ⊂ PΘ sb0, − and since l ∈ KΘ one has that wb0 ∈ PΘ sb0. By the Bruhat decomposition −1 −1 it follows that WΘw = WΘs so that s ∈ w WΘ. Hence

−1 + −1 + λ(ξ)= s H ∈ w WΘH , which implies the reverse inclusion in (5).

Now we are ready to state the permutation of the Lyapunov and Morse spectra of the Morse components under the Weyl group. Theorem 8.3. For every w ∈ W we have

−1 + −1 + ΛLy (M (w)) = w ΛLy(M ) and ΛMo (M (w)) = w ΛMo(M ). Proof: The statement for the Lyapunov spectra implies that for the Morse spectra because ΛMo (M (w)) is the convex closure of ΛLy (M (w)). For the Lyapunov spectra we recall that any Lyapunov exponent λ is λ = λ (r · b)= λ (sk, b) for a regular point r ∈ R and b ∈ F where sk = S (φk (r)) (see Proposition 6.6 and Corollary 6.7). Hence,

−1 + ΛLy(M(w)) = ΛLy(M(w)x)= w WΘλ (x)= x reg x regular[ [ −1 + −1 + = w WΘλ (x) = w ΛLy(M ), x reg ! [ 33 which proves the theorem.

Remark: The above theorem says that the several Morse spectra are homeomorphic to each other. This is in consonance with the fact that the intersection of the different chain components with the fibers are homeomorphic as well. In fact, a such intersection can be identified to the orbit KHφ wb0, which in turn identifies to the coset space KHφ /M. Combining this theorem with the inclusion in chambers stated in Corol- lary 7.6 we get the following localization result for the Morse (and Lyapunov) spectra.

Corollary 8.4. For each w ∈ W the Lyapunov and Morse spectra ΛLy (M (w)) and ΛMo (M (w)) are contained in the open cone

−1 + int w WΘphi cla . It follows that ΛMo (M (w1)) ∩ ΛMo (M (w2)) = ∅ if M (w1) =6 M (w2), that is, if Wφw1 =6 Wφw2.

−1 + −1 + Proof: In fact, int w WΘ(φ)cla = w int WΘ(φ)cla and the right + hand side contains ΛMo(M ). The last statement follows from the fact that −1 + −1 + the open cones int w1 WΘ(φ)cla and int w2 WΘ(φ)cla are either equal or disjoint.

In particular, the equalities in Theorem 8.3 say that if w ∈ Wφ, that is, + + + + if M (w) = M (1) = M , then wΛLy(M )=ΛLy(M ) and wΛMo(M ) = + ΛMo(M ). This invariance is easily carried over to the other spectra, by −1 taking a conjugate of Wφ. Namely if s ∈ w Wφw then sΛLy (M (w)) = ΛLy (M (w)) and sΛMo (M (w))=ΛMo (M (w)), as follows directly from the theorem. Actually, we have the following more precise result.

−1 Corollary 8.5. For each w ∈ W we have w Wφw = {s ∈ W : sΛMo (M (w)) = ΛMo (M (w))}. The same statement holds with ΛLy instead of ΛMo.

−1 + Proof: In fact, s ∈ W ﬁxes ΛMo (M (w)) if and only if wsw ﬁxes ΛMo (M ). + By the above theorem ΛMo (M ) is invariant under Wφ. Conversely, if + + −1 + σΛMo (M )=ΛMo (M ) then ΛMo (M (σ )) = ΛMo (M ) hence σ ∈ Wφ.

34 In view of this corollary the parabolic subgroup Wφ can be recovered from the Morse spectra of φ (more speciﬁcally from the Morse spectrum + ΛMo (M )). In other words the parabolic type of φ is completely determined by its spectra. Another way of expressing the relationship between the exponents and the parabolic type is by comparing Θ(φ) with the set of simple roots annihilating some Morse exponents, as stated in the next consequence of Theorem 8.3.

+ Corollary 8.6. For a Morse exponent λ ∈ ΛMo (M ) write Θ(λ) = {α ∈ + Σ: α (λ)=0}. Then Θ(λ) ⊂ Θ(φ). Also there exists λ0 ∈ ΛMo (M ) such that Θ(λ) ⊂ Θ(φ).

Proof: The inclusion Θ(λ) ⊂ Θ(φ) is just a restatement of Corollary 7.4. + On the other hand the Wφ-invariance of convex set ΛMo (M ) implies this set contains a point λ0 fixed by Wφ. This fixed point satisfies Θ(λ0) ⊃ Θ(φ), hence the equality.

+ In other words the block form of φ is given by the elements in ΛMo (M ) with the smallest possible degree of regularity. Hence, again the block form of φ can be read off from the its Morse spectrum. As an immediate consequence we mention that the parabolic type Θ = Σ (or block form Hφ = 0) corresponds to chain transitivity of the flow on the flag bundles. Hence we have the following criterion for chain transitivity on the flag bundles.

Proposition 8.7. The flow φt is chain transitive in some flag bundle if and + only if 0 ∈ ΛMo(M ). In this case φt is chain transitive (and chain recurrent) on every flag bundle.

Example: If the base X is a point then our flow is just the iteration of a single element g ∈ G (in the discrete time case) or the action of a one- parameter group exp(tX) in G, X ∈ g (continuous time). For the discrete- time case generated by g ∈ G, let g = uhn be its Jordan-Schur decomposition (see [14], Chapter IX), where u is elliptic, h is hyperbolic, and u unipotent. Choose a Weyl chamber A+of G such that h ∈ clA+and put h = exp(H+), H+ ∈ cla+. Then it can be shown that the block form of this flow is the block form of the hyperbolic part H+ (see In [24]). Also the finest Morse decomposition in F is given by the fixed point set of h, that is, M(w) = fix(H+,w). To compute the vector exponents of this flow we fix a Cartan

35 and Iwasawa decomposition of G compatible with the chamber A+. The polar exponent of gk is precisely H+ (cf. [29], Theorem 2.2) . So that by our results it follows that

−1 + ΛLy(M(w))=ΛMo(M(w)) = w H .

9 Representations and vector bundles

In this final section we establish some relationships between the a-Lyapunov and Morse spectra and the exponents of a linear flow on a vector bundle. Let V → X be a finite dimensional real vector bundle with compact Hausdorff base space and φt a linear flow on V, assumed to be chain transitive on the base space. The bundle Q = BV → X of frames of V is a principal bundle with structural group Gl (m, R), m = rank V. The linear flow φt maps frames into frames, hence it lifts to a right invariant flow (also denoted by φt) on BV. Fix in V a Riemannian metric h·, ·i and let OV be the O(n)-subbundle of orthonormal frames of BV. Then OV· AN and OV· S are, respectively, the Iwasawa and Cartan decompositions of BV, where A is the subgroup of diagonal matrices (with positive entries), N is the subgroup of upper triangular unipotent matrices and S the set of positive definite matrices of Gl(m, R). The spectra are defined by the cocycle a (t, r) = log at ∈ a, where r ∈ OV, a is the subspace of diagonal matrices and at is the A-component in the Iwasawa decomposition of φt(r)= rt · atnt ∈ OV· AN. This cocycle descends to the flag bundle FV of the flags of subspaces of V. Write a = R · id ⊕ a, where a is the subspace of zero trace diagonal matrices. Then the component of the spectra in the direction of the scalar matrices R · id is the same for any Morse component in FV (see Section 4.4), while the a-component is invariant under the Weyl group (permutation group of m letters). Now, let PV be the projective bundle of V. The spectra of the flow on V is defined by the cocycle a|·| (t, v) over PV given by

|φ v| a (t, v) = log t v ∈V\{0}. |·| |v|

36 To get the relation between the two cocycles take v ∈ V with |v| = 1 and m write v = r · e1, r ∈ OV and e1 ∈ R the ﬁrst basic vector. Then φt (v) = φt (r) · e1. Take the Iwasawa decomposition φt (r) = rt · atnn ∈ OV· AN. Then |φtv| = |atnte1| = |ate1|. So that

a|·| (t, v)= λ1 (log at) where λ1 : a → R is the ﬁrst eigenvalue λ1 (diag{a1,...,am})= a1. Hence

a|·| (t, v)= λ1 (a (t, ξ)) for any ξ ∈ FV projecting into [v] ∈ PV. On the other hand a chain component on the projective bundle is the projection of chain components on the ﬂag bundle. This implies the following expressions for the spectra of the linear ﬂow on the vector bundle. P |·| Proposition 9.1. Let M ⊂ V be a chain component and denote by ΛMo (M) |·| |·| |·| and ΛLy (M) its spectra. Then ΛMo (M) = λ1 ΛMo M and ΛLy (M) = λ Λ M , where M is any chain component on FV projecting onto M. 1 Ly

In particular if the block form of φt is a regular matrix then there are m!, m = rankV, chain components on the maximal flag bundle FV. Each one meets the fibers in singletons. By the previous results (see Corollary 7.6 and Theorem 8.3) each Morse spectra of a chain component M (w) is entirely contained in the Weyl chamber wa+, and so contains only regular elements. The chain components on FV project down to m components on the projective bundle PV, which are ordered linearly. The vector bundle |·| Morse spectrum ΛMo of the attractor component is just the set of highest + eigenvalues of the matrices in ΛMo (M ). For the other components one + must apply a permutation to the elements of ΛMo (M ) and then take the highest eigenvector. This amounts to take the successive eigenvectors of + + the elements of ΛMo (M ). Since the elements of ΛMo (M ) (and hence of + ΛLy (M )) are regular matrices this yields the simplicity of the Lyapunov spectrum of a nominal trajectory on the base space. We note that the above proposition can be easily extended to obtain the spectra on the Grassmann bundles GrkV from the vector spectra on the flag bundle FV: One needs only to replace PV by GrkV and the functional λ1 by the functional λk which gives the sum of the k-first eigenvalues λk (diag{a1,...,am})= a1 + ··· + ak.

37 In the general case the idea is to start with a principal bundle Q → X with structural group G and take a linear representation ρ : G → Gl(V ) on a vector space V . This representation yields the associated bundle V = Q ×ρ V → X which turns out to be a vector bundle. If φt is a right invariant flow on Q then the induced flow on V is linear. For instance if G is semi-simple and connected then an irreducible repre- ∗ sentation ρµ is given by its highest weight µ ∈ a . Then there is a K invariant a |φtv| inner product on V such that the norm cocycle |·| (t, v) = log |v| is given by a|·| (t, v)= µ (log at) with at the A-component in the Iwasawa decomposition of φt(r) where v = r · Y for a highest weight vector Y ∈ V . + Also, for h ∈ clA , log kρµ(h)k = hµ, log hi (cf. [13], Lemma 4.22). Thus the vector bundle spectra of the linear flow on V can be recovered from the intrinsic spectra on the principal bundle. We do not go here into the details, since it requires a discussion of the chain components on the projective bundle of V, which is not yet available.

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